Geometry and measures, Year 8: areas, units, bearings and bisectors

40 cm²

提示One of the three numbers is not needed.

为什么Area = base × perpendicular height = 8 × 5. The slant side plays no part in the area.

9 cm

提示Undo the multiplication.

为什么Base × 4 = 36, so the base is 36 ÷ 4 = 9 cm. The answer is a length, so the unit is cm, not cm².

Cutting off a triangle and sliding it across makes a rectangle

提示Imagine scissors and one straight snip from a top corner.

为什么Cut a right-angled triangle from one end and move it to the other. The result is a rectangle with the same base and the same height, and no area has been lost.

It gets smaller

提示Its base is unchanged, so think about how tall it now stands.

为什么Area is base × perpendicular height. The base and the slant sides are unchanged, but the shape is now lower, so the area shrinks.

10,000 cm²

提示A metre-square tile is 100 cm along each edge.

为什么1 m² is a square 100 cm by 100 cm, and 100 × 100 = 10,000. The length factor of 100 is squared.

30,000 cm²

提示The scale factor for length gets squared.

为什么Each square metre holds 10,000 cm², so 3 × 10,000 = 30,000 cm².

2,000,000 cm³

提示A metre cube is 100 cm in three directions.

为什么1 m³ = 100 × 100 × 100 = 1,000,000 cm³, so 2 m³ is two million. For volume, the length factor is cubed.

0.5 m²

提示Going to the bigger unit means dividing.

为什么5,000 ÷ 10,000 = 0.5. It takes ten thousand square centimetres to fill one square metre, so 5,000 fills half of one.

Clockwise

提示Start facing north and turn towards east.

为什么A bearing is the angle turned from north, in that direction, to face where you are heading. It is written with three figures.

090°

提示A quarter turn from north, written with three figures.

为什么East is 90° round from north. Bearings always have three figures, so a zero is put in front.

250°

提示Coming back means a half turn.

为什么The return direction is exactly opposite, so add 180°: 70 + 180 = 250.

020°

提示Half a turn back, and keep three figures.

为什么Adding 180° would go past 360°, so subtract: 200 − 180 = 20, written as 020°.

The perpendicular bisector

提示Its two words mean 'at 90°' and 'cuts in two'.

为什么It is constructed by drawing arcs of equal radius from each end of the segment and joining the two points where the arcs cross.

It is the same distance from A and B

提示Think about where you could stand to be fair to both ends.

为什么That is why the construction works: the arcs have equal radii, so the points where they cross are equally far from A and from B.

The two arms of the angle

提示A ball rolling along it would stay midway between two walls.

为什么The bisector cuts the angle into two equal angles, so it runs exactly midway between the two lines that form it.

No — leave them as evidence of the method

提示They show how the line was found.

为什么Construction arcs prove that the line was constructed and not measured or guessed. Without them the method cannot be seen.